“The first amendment delays the president’s health care law by one year. And the second permanently repeals Obamacare’s medical device tax that is sending jobs overseas," Boehner said in a joint statement with House Republican leaders following a conference meeting Saturday. "We will also vote on a measure that ensures our troops get paid, no matter what."

Sen. Ted Cruz, R-Texas, who led the push to use the CR as a mechanism for defunding Obamacare, hailed the strategy as a "terrific idea" during an interview Friday evening.

"Existing federal law gives President Obama all the authority he needs to fund the military even if there is a shutdown," Cruz told Fox News' Greta Van Susteren.

"But the Democrats are threatening that, and so if they pass a simple CR that funds the military, sends it over to the Senate, I think it's very, very difficult for Harry Reid to refuse to fund the military. Don't mention Obamacare. Should we fund the military, yes or no? I don't know that you get Democrats to vote no on that on a clean vote and one at a time, one after the other after the other."

Susteren pointed out that the idea originated with Cruz, who told Fox's Neil Cavuto on Sept. 20 that House Republicans should "send over limited continuing resolutions, funding different parts of the government, one at a time."

Doing so would force Senate Democrats into choosing between casting a series of votes that ultimately fund all of government but Obamacare or voting against a series measures enjoying popular support and appearing to want the shutdown to be as painful as possible.